Do Rewards Always Balance Risks in the Casino?

Most people think that, carved in the cosmic book of rules, is some fine print ordaining "thou shalt balance risks with rewards." Those who take small chances should only expect success to bring modest returns. Those willing to battle long odds may presume victory will yield correspondingly high spoils.

Do casinos, where risk and reward are sum and substance of the agony and ecstasy, follow this fiat? Yes... to an extent.

Take roulette. On bets like black, risk $1 and odds are nearly 1?to-1 you'll win an equal amount. On a single number, you fight 37-to-1 odds but your $1 can win $35. Craps is similar. Drop $5 on Pass and odds are close to 1-to-1 you'll win $5. Toss the $5 on the eleven and you're a 17-to-1 underdog to collect $75.

The rule doesn't hold where the universe is warped by slots. What you can win on a try may not offset the difficulty of doing so.

I'll illustrate with deuces-wild video poker. The actual returns shown below yield roughly 100 percent payback. That is, over many bettors, all wagers revert to players and the house breaks even. Machines on which casinos earn a profit are similar, but returns are shaved here or there; for instance, at 4-for-1 rather than 5-for-1 on four-of-a-kind, the house would have 5 percent edge.

hand	actual returns
royal, no deuces	800
four deuces	200
royal with deuces	25
five of a kind	15
straight flush	9
four-of-a-kind	5
full house	3
flush	2
straight	2
three-of-a-kind	1

With these payoffs, individual hits don't match the odds bettors overcome in obtaining the hands. Ideal returns, which precisely balance the risks for optimal play at this game, are as follows:

hand	ideal returns
royal, no deuces	2562.45
four deuces	276.84
royal with deuces	29.82
five of a kind	18.25
straight flush	13.54
four-of-a-kind	1.85
full house	3.58
flush	5.05
straight	1.93
three-of-a-kind	1.19

Comparing actual and ideal returns for various hands reveals two factors of interest.

First, all returns but one on this machine undercompensate for the odds against the corresponding hands. Solid citizens are shorted most strongly on natural royal jackpots, with 800-for-1 instead of the ideal 2562.45-for-1. The aggregate deficit is equalized by 5-for-1 rather than 1.85-for-1 on four-of-a-kind.

Second, four-of-a-kind, full house, and flush actually return $5, $3, and $2 per dollar bet - decreasing amounts which reflect, albeit inaccurately, standard "poker" rankings of these hands. However, based on chances of the various final results when video poker is played according to the optimal strategy for this game, returns should be least for four-of-a-kind and most for flushes.

This is because video poker strategies depend on pay tables and may favor different results than winner-take-all poker pots. As one example, say you were dealt 3-H, 6-H, and three deuces - a straight flush. Deuces are wild. In live five-card draw, you'd normally stand pat. In video poker, you'd usually discard the 3-H and 6-H. The worst that could happen would be to finish with four-of-a-kind, winning 5-for-1 rather than 9-for-1. However, you'd have roughly 4 percent chance of winning 200-for-1 by drawing the last deuce - and could also retain your 9-for-1 straight flush or improve to a 25-for-1 three-deuce royal.

Casinos aren't being naughty by adjusting payoffs. Averages work out as they should. Overpaying the more common hands maximizes playing time at the machines. Which is what everybody wants. Except, of course, when they hit the jackpot - and suddenly join the elite who believe they should get every penny they deserve. Sumner A Ingmark, muse of the machines, said it like this:

On games I choose, here's what I expect,
Less when I lose, more when I collect.